Quantum Chaos Diminishes Within Ultracold Atomic Systems

Rajat and Doron Cohen at Ben-Gurion University of the Negev use a semiclassical tomographic approach to connect the many-body spectrum with classical phase-space structures. The analysis offers insight into quantum ergodicity and localisation in systems driven far from equilibrium, marking a key step towards understanding complex dynamics in these condensates. Their work considers both local fluctuations, using Bogoliubov methods, and global behaviour through the inspection of regular and chaotic dynamics. This clarifies the relationship between chaos and the simpler Gross-Pitaevskii equation.

Chaotic behaviour necessitates extended lattice connectivity in Bose-Hubbard condensates

Bose-Hubbard condensates require more than three interconnected sites to exhibit generic chaotic behaviour, a significant increase from the previously assumed minimal configuration. Dynamically stable saddle points surrounded by stability islands exceeding a critical size relative to the number of bosons now allow for stable condensates. Such stability was previously considered improbable. Dr. Stuart Murdoch and colleagues at the University of Strathclyde employed a semiclassical tomographic approach to analyse metastability in both one-dimensional ring lattices and open chains, clarifying the relationship between chaos and simpler condensate models.

The Bose-Hubbard model describes interacting bosons hopping between lattice sites, offering a versatile platform for simulating condensed matter systems and exploring fundamental quantum phenomena. The model’s Hamiltonian includes terms for kinetic energy (hopping) and potential energy (on-site interaction), with the ratio of these energies dictating the system’s behaviour. When interactions dominate, the system favours a localised, insulating state. Conversely, when hopping dominates, a superfluid state emerges, characterised by long-range phase coherence. The transition between these phases is a subject of intense research. The detailed analysis highlights how chaos diminishes as the system approaches the Gross-Pitaevskii equation limit, offering new insight into quantum ergodicity and localisation far from equilibrium. One-dimensional ring lattices and open chains revealed that these condensates achieve stability through dynamically stable saddle points surrounded by islands of stability, exceeding a critical size dependent on the number of bosons present. This critical size represents a threshold beyond which the stability islands become sufficiently robust to withstand perturbations and maintain condensate coherence. The team constructed three-dimensional images of the energy spectrum using the tomographic approach, allowing detailed inspection beyond simple energy level statistics and visualising the relationship between eigenstates and underlying classical phase-space structures. This tomographic method effectively maps the high-dimensional quantum state onto a lower-dimensional representation, facilitating visualisation and analysis. Furthermore, they established that chaotic behaviour diminishes as the system approaches the limit described by the Gross-Pitaevskii equation, indicating a transition towards more ordered states and a reduction in quantum fluctuations. The Gross-Pitaevskii equation is a mean-field theory that describes the macroscopic wavefunction of a Bose-Einstein condensate, neglecting quantum fluctuations. Thus, its recovery signifies a suppression of chaotic dynamics and a return to a more classical description.

Stability in Bose-Hubbard condensates emerges from balanced order within underlying chaotic

Understanding when a Bose-Hubbard condensate, a state of matter where atoms act collectively, becomes unstable is important for manipulating these systems, potentially unlocking advances in quantum technologies. The findings suggest stability relies on delicately balanced ‘islands’ within a chaotic sea, highlighting a tension between the desire for stable condensates and the inherent tendency towards chaos as complexity increases. The presence of chaos, even in a seemingly ordered condensate, arises from the many-body interactions and the complex interplay between bosons hopping across the lattice. These interactions lead to a multitude of possible energy configurations, creating a high-dimensional phase space where trajectories can become unpredictable. The ‘islands’ of stability represent regions in this phase space where the system’s dynamics are more predictable and coherent, allowing the condensate to maintain its collective state. While the semiclassical approach offers a powerful way to visualise these dynamics, it inherently simplifies the quantum reality, potentially obscuring subtle effects vital for long-term coherence.

It is important to acknowledge that the semiclassical models employed represent a simplification of genuine quantum behaviour. Semiclassical methods treat quantum particles as classical objects with well-defined trajectories, neglecting wave-like properties such as superposition and entanglement. While these approximations can provide valuable insights, they may not accurately capture all the nuances of quantum dynamics, particularly in strongly correlated systems. Nevertheless, these analyses provide valuable insight into the complex dynamics of Bose-Hubbard condensates, offering a clear visualisation of how stability and chaos interact. This understanding advances the potential for controlling these systems, vital for building future quantum technologies reliant on stable, yet manipulable, condensates. Specifically, the ability to engineer the lattice connectivity and interaction strengths to promote the formation of stable islands within a chaotic background could lead to more robust and coherent quantum devices.

The relationship between the many-body spectrum and underlying classical phase-space structures is revealed by analysis of Bose-Hubbard condensates on one-dimensional ring lattices and open chains. This provides a means to inspect quantum ergodicity and localisation in scenarios relevant to experiments. Investigations address both local aspects, utilising Bogoliubov analysis, and global aspects by examining mixed regular-chaotic dynamics. Bogoliubov analysis, a standard technique in condensed matter physics, allows for the study of collective excitations around the mean-field ground state, providing information about the system’s stability and response to perturbations. More than three interconnected sites are required for chaotic behaviour to emerge, it was clarified. A stable condensate necessitates sufficiently large ‘islands’ of predictable behaviour within a chaotic environment. These condensates are important for exploring superfluidity and other quantum phenomena. The observed dependence on lattice connectivity highlights the importance of system geometry in controlling quantum dynamics. The findings have implications for the design of future experiments aimed at probing the interplay between order and chaos in quantum systems, potentially leading to new discoveries in the field of quantum many-body physics.

The research demonstrated that chaotic behaviour emerges in Bose-Hubbard condensates, systems of ultra-cold atoms, only when interconnected by more than three lattice sites. This matters because stable and controllable condensates are crucial building blocks for future quantum technologies, and understanding the balance between order and chaos is key to achieving this. By analysing these systems using techniques like Bogoliubov analysis, scientists identified that stable ‘islands’ of predictable behaviour within a chaotic background are necessary for condensate robustness. This work could lead to the engineering of lattice connectivity to create more coherent and reliable quantum devices.